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Silicene consists of a monolayer of silicon atoms in a buckled honeycomb structure. It was recently discovered that the symmetry of such a system allows for interesting Rashba spin–orbit effects. A perpendicular electric field is able to couple to the sublattice pseudospin, making it possible to electrically tune and close the band gap. Therefore, external electric fields may generate a topological phase transition from a topological insulator to a normal insulator (or semimetal) and vice versa. The contribution of the present paper to the study of silicene is twofold. Firstly, we perform a group theoretical analysis to systematically construct the Hamiltonian in the vicinity of the K points of the Brillouin zone and find an additional, electric field induced spin–orbit term, that is allowed by symmetry. Subsequently, we identify a tight-binding model that corresponds to the group theoretically derived Hamiltonian near the K points. Secondly, we start from this tight-binding model to analyze the topological phase diagram of silicene by an explicit calculation of the ℤ2 topological invariant of the band structure. To this end, we calculate the ℤ2 topological invariant of the honeycomb lattice in a manifestly gauge invariant way which allows us to include Sz symmetry breaking terms—like Rashba spin–orbit interaction—into the topological analysis. Interestingly, we find that the interplay of a Rashba and an intrinsic spin–orbit term can generate a non-trivial quantum spin Hall phase in silicene. This is in sharp contrast to the more extensively studied honeycomb system graphene where Rashba spin–orbit interaction is known to compete with the quantum spin Hall effect in a detrimental way. [ABSTRACT FROM AUTHOR]

We propose a scheme of band engineering by means of staggered electric potential, antiferromagnetic exchange field and intrinsic spin-orbit coupling for electrons on a honeycomb lattice. With fine control on the degrees of freedom of spin, sublattice and valley, one can achieve a topological state with simultaneous non-zero charge and spin Chern numbers. With first principles calculations, we confirm that the scheme can be realized by material modification in perovskite G-type antiferromagnetic insulators grown along the [111] direction, where Dirac electrons from d orbits are achieved on an atomic sheet of a buckled honeycomb lattice. In a finite sample, this state provides a spin-polarized quantum edge current, robust to both non-magnetic and magnetic defects, with the spin polarization reversible by electric field, and is thus ideal for spintronics applications. [ABSTRACT FROM AUTHOR]